We consider the two-dimensional Rayeigh-B\'enard convection problem between Navier-slip fixed-temperature boundary conditions and present a new upper bound for the Nusselt number. The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result in Whitehead & Doering (2011) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (2022) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number and slip-length, our upper bounds indicate five possible scaling laws: $\textit{ Nu} \sim (L_s^{ -1}\textit{ Ra})^{ \frac{ 1}{ 3}}$, $\textit{ Nu} \sim (L_s^{ -\frac{ 2}{ 5}}\textit{ Ra})^{ \frac{ 5}{ 13}}$, $\textit{ Nu} \sim \textit{ Ra}^{ \frac{ 5}{ 12}}$, $\textit{ Nu} \sim \textit{ Pr}^{ -\frac{ 1}{ 6}} (L_s^{ -\frac{ 4}{ 3}}\textit{ Ra})^{ \frac{ 1}{ 2}}$ and $\textit{ Nu} \sim \textit{ Pr}^{ -\frac{ 1}{ 6}} (L_s^{ -\frac{ 1}{ 3}}\textit{ Ra})^{ \frac{ 1}{ 2}}$
Categories: math.AP math-ph math.MP physics.flu-dyn